A fast direct solver for the biharmonic problem in a rectangular grid

Matania Ben-Artzi*, Jean Pierre Croisille, Dalia Fishelov

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

40 Scopus citations

Abstract

We present a fast direct solver methodology for the Dirichlet biharmonic problem in a rectangle. The solver is applicable in the case of the second order Stephenson scheme [J. W. Stephenson, J. Comput. Phys., 55 (1984), pp. 65-80] as well as in the case of a new fourth order scheme, which is discussed in this paper and is based on the capacitance matrix method ([B. L. Buzbee and F. W. Dorr, SIAM J. Numer. Anal., 11 (1974), pp. 1136-1150], [P. Bjørstad, SIAM J. Numer. Anal., 20 (1983), pp. 59-71]). The discrete biharmonic operator is decomposed into two components. The first is a diagonal operator in the eigenfunction basis of the Laplacian, to which the FFT algorithm is applied. The second is a low-rank perturbation operator (given by the capacitance matrix), which is due to the deviation of the discrete operators from diagonal form. The Sherman-Morrison formula [G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., John Hopkins University Press, Baltimore, MD, 1996] is applied to obtain a fast solution of the resulting linear system of equations.

Original languageEnglish
Pages (from-to)303-333
Number of pages31
JournalSIAM Journal on Scientific Computing
Volume31
Issue number1
DOIs
StatePublished - 2008

Keywords

  • Biharmonic problem
  • Capacitance matrix method
  • Compact scheme
  • Driven cavity
  • Fast Fourier transform
  • Fast solver
  • Navier-Stokes equations
  • Sherman-Morrison formula
  • Stephenson scheme
  • Streamfunction formulation
  • Vorticity

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